Straight-line graph

Quite often isochronous data is presented on log-log scales. One of the reasons for this is that on linear scales any slight, but possibly important, non-linearity between stress and strain may go unnoticed whereas the use of log-log scales will usually give a straight-line graph, the slope of which is an indication of the linearity of the material. If it is perfectly linear the slope will be 45°. If the material is non-linear the slope will be less than this. 

The straight line graph of stress against log (time) confirms the relationship... 

If the kinetics of the reaction disobey the Michaelis-Menten equation, the violation is revealed by a departure from linearity in these straight-line graphs. We shall see in the next chapter that such deviations from linearity are characteristic of the kinetics of regulatory enzymes known as allosteric enzymes. Such regulatory enzymes are very important in the overall control of metabolic pathways. 

Charles and Gay-Lussac carried out a number of experiments with the hope of improving the performance of their balloons. They found that, provided the pressure is kept constant, the volume of a gas increases as its temperature is raised. In this case, a straight-line graph is obtained when the volume is plotted against the temperature (Fig. 4.10). This result implies... 

The formula of a straight-line graph of y (the vertical axis) plotted against x (the horizontal axis) is... 

Many of the equations that we meet in the text can be rearranged to give a straight-line graph when plotted as shown in the following table. 

An intuitive way to measure the rate constant, k, is to prepare some number of molecules in the excited state, perhaps with a pulse of light, and observe the amount of light given off per unit of time after the pulse. A logarithmic plot of intensity with time will give a straight-line graph with a slope of -k. This approach is the basis of time domain measurements. 

For the straight-line graph of Rate versus [Enzyme], an excess of substrate must be present. 

Figure 5.19 Arrhenius plot of diffusion data, In D versus 1/T. The slope of the straight-line graph allows the activation energy of diffusion, a, to be determined, and the intercept at 1/T = 0 gives a value for the pre-exponential factor.

Figure S5.1 Straight-line graph of In c versus x2 from a diffusion experiment The slope of the graph allows a value for the diffusion coefficient, D, to be determined.

Another problem with real data is that due to random indeterminate errors (Chapter 1), the analyst cannot expect the measured points to fit a straight-line graph exactly. Thus it is often true that we draw the best straight line that can be drawn through a set of data points and the unknown is determined from this line. A linear regression, or least squares, procedure is then done to obtain the correct position of the line and therefore the correct slope, etc. 

A plot of the reciprocal of velocity against reciprocal of the substrate concentration gives a straight line graph with intercepts of... 

Cox chart chem A straight-line graph of the logarithm of vapor pressure against a special nonuniform temperature scale vapor pressure-temperature lines for many substances intersect at a common point on the Cox chart. kaks, chart cp See chemically pure. 

Note T Urbanski ( ). Rychter, CR 208, 900(1939) found for eleven expls tested by them that logarithmic relationship between the time lag prior to expln and the reciprocal of absolute temperature gives a straight line. Graphs for NC s, MF, ErTeN and EDNA did not give straight lines... 

Generally, the energy of the olefin n -level is more sensitive to substituent effects than the 7r-level (see Table IV for example). Therefore, often a simple plot of the equilibrium constant for the metal-olefin complex vs the energy of the lowest unoccupied olefin molecular orbitai gives a reasonable straight-line graph. Such a plot reported by Tolman (220) for the equilibrium... 

Whenever possible, cast data into such a form that a straight-line graph results from their plotting. A straight line is much easier to draw accurately than a curved one often one can obtain important information from the slope or intercept of the straight line. If the two variables under discussion are x and y (the convention is to plot x as the abscissa andy as the ordinate), and if they are linearly related (i.e., if the graph is a straight line), the form of the mathematical equation that represents this line is... 

From this expression, it follows that a straight-line graph should be obtained if we let y = log P and x = IT then the equation will be of the form... 

Figure 6-5 shows the plot of log P against l/7 The slope of this straight-line graph is... 

If you determine the rate constant for a reaction (as in the first part of this chapter) at several different temperatures, you can plot your data as log k versus 1 IT (Figure 15-7) and obtain a straight-line graph, preferably by the method of least squares. 

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